Permutation help on test review?

Question

What is the smallest positive integer, k, such that p^k is the identity function, where p is the permutation (2345)(1678910) in S (sub 10)? Explain briefly.

Answer 1

There are two distinct cycles here, one of length four and a second of length six. Applying p^k where k is any multiple of four will leave the elements in the first cycle unchanged, and similarly applying p^k where k is a multiple of 6 will leave the elements in the second cycle unchanged. Therefore the smallest k which will leave all the elements unchanges is simply the least common multiple of 4 and 6, which is 12. Ergo, k=12